As part of our attack on the protein-folding problem, we have been using simple statistical-mechanical models to study the folding properties of single-domain proteins. In these models it is assumed that the folding properties of a protein are determined entirely by the intramolecular interactions present in its native folded conformation. The backbone structure of polypeptide chains is described in terms of the conformations of individual structural units which are identified with pairs of flexible dihedral angles. These units are viewed as having only two possible conformational states, the "native" state, corresponding to the conformation assumed by the unit in the native structure of the chain, and the "random-coil" state, which encompasses all non-native conformations. The structural states of a polypeptide chain are then defined by the sequence of native/random-coil states of these structural units. The stability of any such state is determined by the offsetting effects of entropy loss associated with fixing dihedral-angle pairs in the native conformation, and native non-bonding contacts between different parts of the chain. For a given chain these stabilizing contacts are identified by analyzing available X-ray crystallographic or NMR-derived structures of the corresponding protein. In our initial studies, the intra-chain contacts in a state could only connect parts of the chain separated by sufficiently long contiguous stretches of native dihedral-angle pairs. We have generalized this description to include the entropy of "loops" created by contacts between parts of the chain separated by stretches of non-native dihedral-angle pairs. Specified values for the entropy loss of fixing a single unit in the native conformation and for the overall strength of the intra-chain contacts are then used to compute a model partition function of the chain. The number of states that arises from complete enumeration of the possible combinations of dihedral-angle-pair states becomes intractably large for even a moderately long chain. However, we have developed methods for computing exact partition functions for model polypeptide chains of arbitrary length in certain cases. For the remaining cases we compute the partition function using the "single-sequence", "double-sequence", and "triple-sequence" approximations--i.e., only considering states which have at most one, two, or three contiguous stretches of native dihedral-angle pairs, respectively. These partition functions are used to compute the free energy for a given chain as a function of a single reaction coordinate defined for each state as either the total number of native dihedral-angle pairs or the fraction of native contacts formed in that state; this "reaction free-energy surface" provides the basis for modeling the equilibrium and kinetic folding properties of the chain. We have developed a procedure for "combinatorial modeling", which allows us to efficiently screen a very broad class of models, incorporating a wide variety of model assumptions--e.g., single-sequence vs. double-sequence vs. exact partition-function calculation, different choices of reaction coordinate, different criteria for identifying intra-chain contacts, etc. The 76 possible combinations of these model assumptions have been used to analyze the equilibrium and kinetic folding properties of a set of more than 20 two-state proteins. This approach has allowed us to identify certain model assumptions that consistently produce more accurate descriptions of the measured folding properties of these proteins. One intriguing observation is that model descriptions incorporating a "coarse-grained" enumeration of native contacts based only on the structure of the alpha-carbon backbone of the protein perform as well as or better than model descriptions using native contacts based on a more detailed picture of the protein structure. We are currently using the combinatoorial approach to perform a key test of the ability of these simple statistical-mechanical models to predict changes in folding properties of individual proteins caused by mutations. In keeping with our interest in developing and exploiting new methods for probing the structural dynamics of macromolecules, we are collaborating with Philip Anfinrud's group in the Laboratory of Chemical Physics in the development of new algorithms for the analysis of Laue (i.e., polychromatic illumination) diffraction data acquired in time-resolved X-ray crystallographic studies of proteins. These algorithms include efficient procedures for the assignment of Miller indices to observed reflections, the integration of intensities to produce structure factors for these reflections, and the scaling of the resulting sets of structure factors from multiple images onto a common intensity scale.